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%%%%% Auteur
\author{\firstname{Nicholas} \lastname{Proudfoot}}
\address{University of Oregon\\
Department of Mathematics\\
Eugene\\
OR 97403, USA}
\email{njp@uoregon.edu}
\thanks{We thank Tom Braden for his feedback on a preliminary draft of this work. The author is supported by NSF grant DMS-1954050.}
%%%%% Sujet
\keywords{Incidence algebra, Kazhdan--Lusztig--Stanley polynomial, matroid.}
\subjclass{05E18, 05B35}
%%%%% Gestion
\DOI{10.5802/alco.174}
\datereceived{2020-09-14}
\daterevised{2021-04-04}
\dateaccepted{2021-04-04}
%%%%% Titre et résumé
%% The title of the paper: amsart's syntax.
\title{Equivariant incidence algebras and equivariant Kazhdan--Lusztig--Stanley theory}
\begin{abstract}
We establish a formalism for working with incidence algebras of posets with symmetries, and we develop
equivariant Kazhdan--Lusztig--Stanley theory within this formalism. This gives a new way of thinking about
the equivariant Kazhdan--Lusztig polynomial and equivariant $Z$-polynomial of a matroid.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\datepublished{2021-08-17}
\begin{document}
\maketitle
\section{Introduction}
The incidence algebra of a locally finite poset was first introduced by Rota, and has proved to be a natural
formalism for studying such notions as M{\"o}bius inversion~\cite{Rota-incidence}, generating functions~\cite{incidence-generating},
and Kazhdan--Lusztig--Stanley polynomials~\cite[Section~6]{Stanley-h}.
A special class of Kazhdan--Lusztig--Stanley polynomials that have received a lot of attention recently is that of Kazhdan--Lusztig polynomials
of matroids, where the relevant poset is the lattice of flats~\cite{EPW,KLS}. If a finite group $W$ acts on a matroid $M$ (and therefore on the lattice of flats),
one can define the $W$-equivariant Kazhdan--Lusztig polynomial of $M$~\cite{GPY}.
This is a polynomial whose coefficients are virtual representations of $W$, and has the property that taking dimensions recovers
the ordinary Kazhdan--Lusztig polynomial of $M$. In the case of the uniform matroid of rank $d$ on $n$ elements, it is actually
much easier to describe the $S_n$-equivariant Kazhdan--Lusztig polynomial, which admits a nice description in terms of partitions of $n$,
than it is to describe the non-equivariant Kazhdan--Lusztig polynomial~\cite[Theorem~3.1]{GPY}.
While the definition of Kazhdan--Lusztig--Stanley polynomials is greatly clarified by the language of incidence algebras,
the definition of the equivariant Kazhdan--Lusztig polynomial of a matroid is completely {\em ad hoc} and not nearly as elegant.
The purpose of this note is to define the equivariant incidence algebra of a poset with a finite group of symmetries, and to show
that the basic constructions of Kazhdan--Lusztig--Stanley theory make sense in this more general setting.
In the case of a matroid, we show that this approach recovers the same equivariant Kazhdan--Lusztig polynomials that were defined
in~\cite{GPY}.
\section{The equivariant incidence algebra}
Fix once and for all a field $k$.
Let $P$ be a locally finite poset equipped with the action of a finite group $W$.
We consider the category $\CWP$ whose objects consist of
\begin{itemize}
\item a $k$-vector space $V$
\item a direct product decomposition $V = \prod_{x\leq y\in P} V_{xy}$, each $V_{xy}$ finite dimensional
\item an action of $W$ on $V$ compatible with the decomposition.
\end{itemize}
More concretely, for any $\sigma\in W$ and any $x\leq y\in P$, we have a linear map
\[
\varphi^{\sigma}_{xy}:V_{xy}\to V_{\sigma(x)\sigma(y)},
\]
and we require that $\varphi^{e}_{xy} = \id_{V_{xy}}$ and that
$\varphi^{\sigma'}_{\sigma(x)\sigma(y)}\circ\varphi^{\sigma}_{xy} = \varphi^{\sigma'\sigma}_{xy}$.
Morphisms in $\CWP$ are defined to be linear maps that are compatible with both the decomposition and the action.
This category admits a monoidal structure, with tensor product given by
\[
(U\otimes V)_{xz} \coloneqq \bigoplus_{x\leq y\leq z}U_{xy}\otimes V_{yz}.
\]
Let $\IWP$ be the Grothendieck ring of $\CWP$; we call $\IWP$ the \emph{equivariant incidence algebra} of $P$
with respect to the action of $W$.
\begin{exam}
If $W$ is the trivial group, then $\IWP$ is isomorphic to the usual incidence algebra of $P$ with coefficients in $\Z$.
That is, it is isomorphic as an abelian group to a direct product of copies of $\Z$, one for each interval in $P$, and multiplication
is given by convolution.
\end{exam}
\begin{rema}\label{base change}
If $W$ acts on $P$ and $\psi:W'\to W$ is a group homomorphism, then $\psi$ induces a functor $F_\psi:\CWP\to\cC^{W'}\!(P)$
and a homomorphism $R_\psi:\IWP\to I^{W'}\!(P)$.
\end{rema}
We now give a second, more down to earth description of $\IWP$.
Let $\VRep(W)$ denote the ring of finite dimensional virtual representations of $W$ over the field $k$.
A group homomorphism $\psi:W'\to W$ induces a ring homomorphism
\[
\Lambda_\psi:\VRep(W)\to\VRep(W').
\]
For any $x\in P$, let $W_x\subset W$ be the stabilizer of $x$. We also define
$W_{xy} \coloneqq W_x\cap W_y$ and $W_{xyz} \coloneqq W_x \cap W_y\cap W_z$.
Note that, for any $x,y\in P$ and $\sigma\in W$, conjugation by $\sigma$ gives a group isomorphism
\[
\psi_{xy}^\sigma:W_{xy}\to W_{\sigma(x)\sigma(y)},
\]
which induces a ring isomorphism
\[
\Lambda_{\psi_{xy}^\sigma}:\VRep(W_{\sigma(x)\sigma(y)})\to \VRep(W_{xy}).
\]
An element $f\in \IWP$ is uniquely determined by a collection
\[
\{f_{xy}\mid x\leq y\in P\},
\]
where $f_{xy}\in\VRep(W_{xy})$
and for any $\sigma\in W$ and $x\leq y\in P$, $f_{xy} = \Lambda_{\psi_{xy}^\sigma}\left(f_{\sigma(x)\sigma(y)}\right)$.
The unit $\delta\in\IWP$ is characterized by the property that $\delta_{xx}$ is the 1-dimensional trivial representation of $W_x$ for all $x\in P$
and $\delta_{xy} = 0$ for all $x 0$ if $x